BATHTUB-TYPE CURVES IN RELIABILITY AND BEYOND

Finding optimal, or at least good, maintenance and repair policies is crucial in reliability engineering. Likewise, describing life phases of human mortality is important when determining social policy or insurance premiums. In these tasks, one searches for distributions to fit data and then makes inferences about the population(s). In the present paper, we focus on bathtub‐type distributions and provide a view of certain problems, methods and solutions, and a few challenges, that can be encountered in reliability engineering, survival analysis, demography and actuarial science.     

一种基于加权F-范数的半正定矩阵的逼近方法

正定性是许多金融预测模型的重要假设前提，然而从实际样本中得到的相关系数矩阵并不能保证其正定性。为此在介绍如何根据样本设定相关系数矩阵以及范数逼近原理的基础上，如何根据该原理找到与之最接近的相关系数矩阵，即最接近的单位对角半正定对称矩阵。通过实证，验证了其方法的有效性。     

中国交通运输业发展的实证研究

采用协整和误差纠正模型与方法，对中国改革开放20多年来交通运输发展与一些相关影响因素之间的关系进行实证研究。结果表明：旅客运输需求与国民收入、乘车费用之间，货物运输与国民经济、燃油价格之间分别存在长期稳定关系。研究认为：中国交通的发展应适当超前于国民经济的发展。同时，研究还发现：“旅游黄金周”的实施并不是促进中国旅客运输需求的显著影响因素。     

Double Sampling Ratio-product Estimator of a Finite Population Mean in Sample Surveys

It is well known that two-phase (or double) sampling is of significant use in practice when the population parameter(s) (say, population mean X¯) of the auxiliary variate x is not known. Keeping this in view, we have suggested a class of ratio-product estimators in two-phase sampling with its properties. The asymptotically optimum estimators (AOEs) in the class are identified in two different cases with their variances. Conditions for the proposed estimator to be more efficient than the two-phase sampling ratio, product and mean per unit estimator are investigated. Comparison with single phase sampling is also discussed. An empirical study is carried out to demonstrate the efficiency of the suggested estimator over conventional estimators.     

Bias Reduction through First-order Mean Correction, Bootstrapping and Recursive Mean Adjustment

Standard methods of estimation for autoregressive models are known to be biased in finite samples, which has implications for estimation, hypothesis testing, confidence interval construction and forecasting. Three methods of bias reduction are considered here: first-order bias correction, FOBC, where the total bias is approximated by the O(T^-1) bias; bootstrapping; and recursive mean adjustment, RMA. In addition, we show how first-order bias correction is related to linear bias correction. The practically important case where the AR model includes an unknown linear trend is considered in detail. The fidelity of nominal to actual coverage of confidence intervals is also assessed. A simulation study covers the AR(1) model and a number of extensions based on the empirical AR(p) models fitted by Nelson & Plosser (1982). Overall, which method dominates depends on the criterion adopted: bootstrapping tends to be the best at reducing bias, recursive mean adjustment is best at reducing mean squared error, whilst FOBC does particularly well in maintaining the fidelity of confidence intervals.     

Assessing Effects on Long-term Survival after Early Termination of Randomized Trials

In this paper we present an approach to using historical control data to augment information from a randomized controlled clinical trial, when it is not possible to continue the control regimen to obtain the most reliable and valid assessment of long term treatment effects. Using an adjustment procedure to the historical control data, we investigate a method of estimating the long term survival function for the clinical trial control group and for evaluating the long term treatment effect. The suggested method is simple to interpret, and particularly motivated in clinical trial settings when ethical considerations preclude the long term follow-up of placebo controls. A simulation study reveals that the bias in parameter estimates that arises in the setting of group sequential monitoring will be attenuated when long term historical control information is used in the proposed manner. Data from the first and second National Wilms' Tumor studies are used to illustrate the method.     

Kernel Density Estimation with Generalized Binning

We propose kernel density estimators based on prebinned data. We use generalized binning schemes based on the quantiles points of a certain auxiliary distribution function. Therein the uniform distribution corresponds to usual binning. The statistical accuracy of the resulting kernel estimators is studied, i.e. we derive mean squared error results for the closeness of these estimators to both the true function and the kernel estimator based on the original data set. Our results show the influence of the choice of the auxiliary density on the binned kernel estimators and they reveal that non-uniform binning can be worthwhile.     

Beta-Bernstein Smoothing for Regression Curves with Compact Support

ABSTRACT. The problem of boundary bias is associated with kernel estimation for regression curves with compact support. This paper proposes a simple and uni(r)ed approach for remedying boundary bias in non-parametric regression, without dividing the compact support into interior and boundary areas and without applying explicitly different smoothing treatments separately. The approach uses the beta family of density functions as kernels. The shapes of the kernels vary according to the position where the curve estimate is made. Theyare symmetric at the middle of the support interval, and become more and more asymmetric nearer the boundary points. The kernels never put any weight outside the data support interval, and thus avoid boundary bias. The method is a generalization of classical Bernstein polynomials, one of the earliest methods of statistical smoothing. The proposed estimator has optimal mean integrated squared error at an order of magnitude n ^−4/5, equivalent to that of standard kernel estimators when the curve has an unbounded support.     

Non-parametric estimation of the long-range dependence exponent for Gaussian processes

We consider a class of long-range-dependent Gaussian processes defined in a semiparametric framework. We propose a new estimator of the long-range dependence parameter, based on the integration of the periodogram in two windows. We show that it is asymptotically Gaussian and calculate the rate of convergence. We optimise parameters defining the window function for the minimum mean-square-error criterion. In a Monte-Carlo study, we compare the proposed estimator with previously studied estimators.     

Error reduction in density estimation under shape restrictions

For the problems of nonparametric estimation of nonincreasing and symmetric unimodal density functions with bounded supports we determine the projections of estimates onto the convex families of possible parent densities with respect to the weighted integrated squared error. We also describe the method of approximating the analogous projections onto the respective density classes satisfying some general moment conditions. The method of projections reduces the estimation errors for all possible values of observations of a given finite sample size in a uniformly optimal way and provides estimates sharing the properties of the parent densities.